4.1 A Frame-Maker’s Puzzle

The term quadrilateral comes from the Latin quadri (“four”) and latus (“side”), a closed shape built from four straight segments. Sketch five random four-cornered doodles and only some will qualify: a quadrilateral must be closed, must use exactly four line segments, and may have no crossings and no curved edges. Its angles are the openings between consecutive sides at the corners.

We open with the two quadrilaterals you know best, the rectangle and the square, through a problem from a metal workshop.

A Puzzle to Begin

A frame-maker wants to cross two slim metal rods so that a wire wrapped around their four tips pulls into a rectangle. The rods will serve as the two diagonals of that rectangle. One rod she already has is 10\ \text{cm} long. Three decisions settle the entire design:

  1. How long must the second rod be?
  2. At what point should the rods be fastened together?
  3. What angle should the rods make at their crossing?
Two metal rods crossed as diagonals, with a wire pulled around their four tips forming a rectangle ? A B C D O rod 1 = 10 cm rod 2 = ? wire pulled tight around the four tips
Figure 4.1 · The frame-maker’s puzzle: two crossed rods as diagonals, a wire pulled tight around their four tips.

This is the thread running through the whole chapter: the diagonals govern the shape. Set the diagonals correctly and the rectangle follows on its own. We will answer the three questions by deduction, reasoning that delivers certainty, not mere confidence.